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Numerical Analysis
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Next, the bidiagonal matrix is transformed to a diagonal matrix by an iterative method in which orthogonal transforms are applied from the left and the right of the matrix. Each iteration applies a QR step to Ak−1TAk−1 implicitly without constructing this product. A shifting strategy is used to accelerate the rate of convergence. The left and right transforms are applied in a way that retains the bidiagonal structure of Ak throughout the process. Off-diagonal entries whose magnitude falls below the threshold are set to zero.
Totally Positive and Totally Nonnegative Matrices
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
An elementary bidiagonal matrix is an n × n matrix whose main diagonal entries are all equal to one, and there is, at most, one nonzero off-diagonal entry and this entry must occur on the super- or subdiagonal. The lower elementary bidiagonal matrix whose elements are given by cij={1,ifi=j,μ,ifi=k,j=k−1,0,otherwise
The Full Navier-Stokes Equations
Published in G. K. Mikhailov, V. Z. Parton, Super- and Hypersonic Aerodynamics and Heat Transfer, 2018
Of the many implicit methods currently being used for solving compressible NS equations, that proposed by Beam and Warming (1976 to 1978) is probably the most popular. Based on the fractional step concept, this is a three-deck, approximately factored, finite-difference, alternate-direction scheme, unconditionally stable and second-order accurate in time; the efficiency of the scheme is ensured by its being constructed for the increments of unknown functions. At each time step in the course of solution, a splitting into three one-dimensional operators is carried out, and a tridiagonal system of difference equations generated by each of these operators is solved by a vector progonka method. This approach is exemplified by the work of Vadyak et al. (1987) on a supersonic aircraft model, and York et al. (1988) and Reddy and Fujiwara (1988) who extended the approach to (viscous) flows with chemical reactions and multicomponent diffusion. A development of the Beam-Warming scheme is the so-called LU factorization approach proposed by the Japanese school (Obayashi and Kuwahara, 1984 and 1986; Fujii and Obayashi, 1985; Obayashi and Fujii, 1988). The approach consists of representing a tridiagonal matrix and a product of two triangle matrices, one with all positive and the other with all negative eigenvalues, and offers the significant computational advantage that instead of a tridiagonal matrix, only a bidiagonal matrix must be inversed. In one further modification of the Beam-Warming approach (Pan and Lomax, 1988), the splitting procedure is required to satisfy the condition that the matrices of the reversed operators be well positioned. An interesting feature of the Pan-Lomax scheme involves the computational stiffness arising from the grid refinement that is needed to incorporate viscous effects into the picture. To eliminate the stiffness, the authors proposed that the error contributed by the largest of the eigenvalues of the coefficient matrix of the problem be excluded from the current solution. Among other implicit alternate-direction schemes, those proposed by Briley and McDonald (1975, 1977) and McDonald and Briley (1975) deserves mention.
Matrix Factorization in Recommender Systems: Algorithms, Applications, and Peculiar Challenges
Published in IETE Journal of Research, 2021
Techniques for computing SVD include, Jacobi techniques, Bidiagonalization techniques, and Bidiagonal SVD [39]. Jacobi techniques are iterative approaches used for computing the singular value decomposition of a matrix. The one sided Jacobi SVD algorithm applies multiplication with Jacobi rotations only to one side of the matrix and it is able to compute singular values with high relative accuracy. The technique that belongs to the category of Bidiagonalization consists of two phases. The first phase utilizes orthogonal transformations to reduce a matrix to a bidiagonal form, while the second phase involves the use of a fast technique for computing the singular value decomposition of a bidiagonal matrix. SVD of the bidiagonal matrix [40] has to be performed to complete the task of computing the singular value decomposition of a general matrix after bidiagonal. Methods for computing bidiagonal SVD include Demmel Kahan bidiagonal SVD and Differential qd algorithms [41].